5.1 Expansion by Minors

Neither of the two previous methods will work for an n by n system if n is larger than three, so we will use another method called expansion by minors. Actually, this method works for any size square matrix, so let's use the same 3 by 3 example with the new method. First, we need to learn some new notation. The real number M_ij is the determinant of a submatrix of dimension n-1 by n-1 which contains everything except row i and column j of the original matrix. The number M_ij is called the minor for element ij of the matrix. For example, if

and we want to find M₁₂, we don't use row 1 or column 2 as shown below.

Therefore, matrix

We will also need something to determine the sign. We set S_ij = (-1)^{i + j} so that s_ij is always either positive one or negative one. For a 4 by 4 matrix,

so s₁₂ is negative one. The cofactor for a_ij is C_ij = s_ijM_ij. This means that C₁₂ = (-1)(-4) = 4 for our example. Now we can write det(A) = a_i1C_i1 + a_i2C_i2 + ... + a_inC_in = a_1jC_1j + a_2jC_2j + ... + a_njC_nj. This means that you take each element of a row or column and multiply it by its cofactor. When you add these terms together, you have the determinant of A. This is a lot easier to see than to explain, so let's find det(A) for

matrix .

Choose any row or column to work with. We will use the first column for the example, but any of them will work. For each position in that column, we will have a_ijs_ijM_ij.

This gives us the same number for the determinant that we found before. Did you notice that by choosing the first column or the second row, we only had to find 2 minors because we know that 0 times anything is 0. Generally, choosing the row or column with the most zeros will save you a lot of work. Now, let's find the determinant of the 4 by 4 matrix,

matrix .

Let's expand along the second row since it contains two zeros.

matrices

Expansion by minors will allow you to find the determinant of a square matrix of any size. However, it requires a lot of operations when the matrix is large because each submatrix used to determine a minor must be expanded. In fact, finding the determinant using this method requires n! operations which is a very large number when n is large because n! = n*(n - 1)*(n - 2)* ... *2*1

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Updated: August 23, 2000